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Theorem bj-eldiag 34586
Description: Characterization of the elements of the diagonal of a Cartesian square. Subsumed by bj-elid6 34580. (Contributed by BJ, 22-Jun-2019.)
Assertion
Ref Expression
bj-eldiag (𝐴𝑉 → (𝐵 ∈ (Id‘𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st𝐵) = (2nd𝐵))))

Proof of Theorem bj-eldiag
StepHypRef Expression
1 bj-diagval2 34585 . . 3 (𝐴𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴)))
21eleq2d 2878 . 2 (𝐴𝑉 → (𝐵 ∈ (Id‘𝐴) ↔ 𝐵 ∈ ( I ∩ (𝐴 × 𝐴))))
3 elin 3900 . . 3 (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (𝐵 ∈ I ∧ 𝐵 ∈ (𝐴 × 𝐴)))
4 ancom 464 . . 3 ((𝐵 ∈ I ∧ 𝐵 ∈ (𝐴 × 𝐴)) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ 𝐵 ∈ I ))
5 bj-elid4 34578 . . . 4 (𝐵 ∈ (𝐴 × 𝐴) → (𝐵 ∈ I ↔ (1st𝐵) = (2nd𝐵)))
65pm5.32i 578 . . 3 ((𝐵 ∈ (𝐴 × 𝐴) ∧ 𝐵 ∈ I ) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st𝐵) = (2nd𝐵)))
73, 4, 63bitri 300 . 2 (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st𝐵) = (2nd𝐵)))
82, 7syl6bb 290 1 (𝐴𝑉 → (𝐵 ∈ (Id‘𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st𝐵) = (2nd𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  cin 3883   I cid 5427   × cxp 5521  cfv 6328  1st c1st 7673  2nd c2nd 7674  Idcdiag2 34582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-iota 6287  df-fun 6330  df-fv 6336  df-1st 7675  df-2nd 7676  df-bj-diag 34583
This theorem is referenced by: (None)
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